The concave bottom extension \(\widehat{V}_a(\rho )\)  is dual to the
convex roof extension \(\widehat{V}_c(\rho )\) . Since the convex roof
extension \(\widehat{V}_c(\rho )\)  is a coherence measure, the concave bottom extension \(\widehat{V}_a(\rho )\)  can be interpreted
operationally in the following way. Suppose Alice holds a state
\(\rho ^A\)  with coherence \(C(\rho ^A)\) . Bob holds another part of the
purified state of \(\rho ^A\) . The joint state between Alice and Bob is
\(\sum _k p_k |\psi _k\rangle _A \otimes |k\rangle _B\)  with \(\rho ^A=\sum _kp_k | \psi _k\rangle \!\langle \psi _k |\) . Bob performs local measurements \(\lbrace |k\rangle \rbrace \) 
and informs Alice the measurement outcomes by classical
communication. Alice's quantum state will be in a pure state
ensemble \(\lbrace  p_k,|\psi _k\rangle \rbrace \)  with average coherence \(\sum _kp_k C(| \psi _k\rangle \!\langle \psi _k |)\) . This process enables Alice to increase the
coherence from \(C(\rho ^A)\)  to the average coherence \(\sum _k p_kC(| \psi _k\rangle \!\langle \psi _k |)\)  due to the convexity of the coherence measure, and it is called the
assisted coherence distillation. The maximum average coherence is
called the coherence of assistance and
quantifies the one way coherence distillation rate [1]}. In this context,
the concave bottom extension \(\widehat{V}_a(\rho )\)  is a coherence of
assistance corresponding to the coherence measure standard
symmetrized variance.
